Emergence of periodic circumferential actin cables from the anisotropic fusion of actin nanoclusters during tubulogenesis

The periodic circumferential cytoskeleton supports various tubular tissues. Radial expansion of the tube lumen causes anisotropic tensile stress, which can be exploited as a geometric cue. However, the molecular machinery linking anisotropy to robust circumferential patterning is poorly understood. Here, we aim to reveal the emergent process of circumferential actin cable formation in a Drosophila tracheal tube. During luminal expansion, sporadic actin nanoclusters emerge and exhibit circumferentially biased motion and fusion. RNAi screening reveals the formin family protein, DAAM, as an essential component responding to tissue anisotropy, and non-muscle myosin II as a component required for nanocluster fusion. An agent-based model simulation suggests that crosslinkers play a crucial role in nanocluster formation and cluster-to-cable transition occurs in response to mechanical anisotropy. Altogether, we propose that an actin nanocluster is an organizational unit that responds to stress in the cortical membrane and builds a higher-order cable structure.

watershed segmentation (#1 -4).Then the actin signals of a single frame before (tn-1) at the potential fusion sites were checked.If the actin signal did not exist at tn-1, the boundary is defined as a newly formed "fusion site" (#1).(b) The fusion site was tracked for the subsequent 11 frames to count a number of frames that actin signals were maintained.In this case, the duration is 4 frames (1.28 sec).absence of motors (a1, a3-a6), and the impact of motors on the cluster formation (a2).The snapshots with black frame are same as the snapshot (1-M) in Fig. 4b in the main text.(b) The simulation results concerning actin stripe formation.The role of the friction asymmetry (b1) and other parameters under the existence of the friction asymmetry (b1-b7) are visualized.The snapshots with black frame are same as the snapshot (4-M) in Fig. 4b in the main text.(c) The interval of the structure estimated by fitting the structure factor S(q) by the gaussian function around the peak.(c1) The clusters, (c2) the labyrinth pattern, (c3) the stripe pattern, and (c4) the clusters for a low filament turnover rate (ω f = 0.01), corresponding to the snapshots (1-M), (3-M), and (4-M) in Fig. 4b and (a4-1) in Supplementary Fig. 7a, respectively.
Supplementary Table 1.Quantification of actin pattern in the fixed embryos at Stages 15 and 16 (Figure 1 e-g, Figure 4c, Supplementary Figure 6a-c) Supplementary Table 2. Quantitative results of the actin pattern of RNAi lines (Figure 3a, Supplementary Figure 3a-c

UAS-lifeact::EGFP(II)
Expresses lifeact fused to GFP under UAS control.Inserted at attP40 site of Chr 2. Generated in this study.

I. COARSE-GRAINED MODEL
We explain our coarse-grained molecular dynamics model of actin filaments, myosin motors, and α-actinin crosslinkers.The model is developed based on our previous model of actin filaments and myosin motors [12], which is in the same spirit as those in many other studies including those in Refs.[13,14].See also the references in Ref. [12] for other related studies of this type of modeling.We extend the model to include crosslinkers.Since the actin nanoclusters and stripes are observed in the vicinity of the apical membrane, we consider the actin cytoskeleton dynamics in a two-dimensional space along the membrane.We further assume a smooth membrane, thus the membranes are not included explicitly.
We start by defining actin filaments, myosin motors, and α-actinin crosslinkers.First, actin filaments are polymer with polarity, and their diameter is about 10 nm and their persistent length is about 1 µm [15].The length of the actin filaments in our experimental system is estimated about 100 nm (Supplement Figure 6a).Based on this estimation, we neglect the bending of actin filament and model a filament by a pair of two discrete particles connected by an elastic spring, which keeps the length of the filament (Fig. 4a in the main text).
The myosin II motor we consider is non-muscle myosin II filament, where a few tens of non-muscle myosin II forming a bipolar bundle with multiple heads on both ends [16].The typical size of myosin II filaments is about 10 nm in width and 200 nm in length.We refer to the myosin II filament simply by motor hereafter.To represent the bipolar structure, we model a motor by a particle with two actin binding domains (ABD) (Fig. 4a in the main text).The two ABD can bind to two different filaments and walk actively along them.
α-actinin form an antiparallel dimer that crosslinks a pair of actin filaments like a motor, but they do not undergo active walking along the filaments unlike the motors [17].The typical size of α-actinin dimer is about 6 nm in width and 36 nm in length.We model α-actinin dimer in the same way as the motor, namely by a particle with two actin binding domains (ABD) (Fig. 4a in the main text).The two ABD bind to a pair of actin filaments, but they do not walk along the actin filaments.To highlight the crosslinking function of the α-actinin dimers, we refer to them as crosslinkers hereafter.
The equations of motion for the filament, motor, and crosslinker particles are given by Here r f f,i represents the position of the ith particle of filament f .Since the filament consists of two particles, i = 1, 2 and the particle i = 0 and 1 correspond to the minus and plus ends, respectively.r m m is the particle position of motor m and r cl c represents the particle position of crosslinker c.The coordinate 0 ≤ α m m,h ≤ 1 along the filament f gives the position of the hth S1).Then, the equation of motion for the motor ABD reads where In the same way, the position of the hth ABD (h = 1, 2) of crosslinker c is specified by the coordinate 0 ≤ α cl c,h ≤ 1 along the filament to which the crosslinker ABD binds.Note that the crosslinker ABD does not walk or slide along the filament to which it binds, and thus its equation of motion does not need to be considered.The filament index f is omitted hereafter for ease of notation.
Inertia terms are neglected because of the small size and velocity of the molecules; The typical sizes of the cytoskeletal filament, the motor, and the crosslinkers are less than submicrometers.The sliding velocity of the motor head is . Schematics explaining the position specified by the coordinate 0 ≤ α ≤ 1 along the filament, and the method to calculate the force f acting at the position α on the particles r f 0 and r f 1 .
submicrometers per second.Note that the filament polarity is taken into account by the order of the filament particle indices, which gives the cue for the motor heads to walk actively towards the filament plus end.The filament, motor, and crosslinker particles experience friction and fluctuation from the surrounding cytosolic fluid, which satisfy the fluctuation-dissipation theorem; Here δ ij is the Kronecker delta that is 1 if i = j and 0 if i = j.The cytosolic friction is defined using a cylindrical approximation [18] as γ f = 3πη cyto (3d f + 2L f 0 /5) and γ m = 6πη cyto (3d m + L m 0 /5) and γ cl = 6πη cyto (3d cl + L cl 0 /5), where η cyto is the cytosolic viscosity and (d f , L f 0 ), (d m , L m 0 ), and (d cl , L cl 0 ) are the width and length of the filament, motor, and crosslinker, respectively.On the other hand, since the motor ABD are bound to and walk along filaments due to the motor walk force f walk , they experience a sliding friction ζ f dα m m,h /dt.The length of the filament is maintained by the stretching elasticity acting between the filament particles: where U harm (r; κ, L 0 ) = (κ/2L 0 )(r − L 0 ) 2 is a harmonic potential with elastic modulus κ and rest length L 0 .
Similarly, the stretching and bending forces act between the motor particles and motor ABD: • r+ is bending energy with bending rigidity k.Here, we use the abbreviations x = |x| and x = x/|x|.Their contribution to the motor ABD, , is assigned to the filament particles with the geometric weight α m m,h , which is necessary because the binding position may be in between the filament particles (Fig. S1).Therefore, the force on the filament particle i from the motors reads Here the summations are calculated over the motor ABD bound to the filament.
Likewise, the length and the straightness of the crosslinkers are maintained by the stretching and bending force acting between the crosslinker particles and crosslinker ABD: where cl c,h = r cl c,h − r cl c .Their contribution to the crosslinker ABD, , is assigned to the filament particles with the geometric weight , where the summations are calculated over the crosslinker ABD bound to the filament.
These treatment of the forces from motors and crosslinkers acting on a filament ensure force and torque conservation of the filament.Note that the counter force of the motor sliding friction and motor walk force, which act on the filaments, vanishes because of the force balance equation (S4).Therefore, Eqs.(S1)-(S4) statistically satisfy the force-and torque-free conditions, which are required for active systems including molecular motors [12] and migrating cells [19,20].See also Ref [12] for details as well as the derivation of Eq. (S4).
In addition to the equations of motion (S1)-(S4), we consider the following stochastic processes.The first one is the process of motor binding and unbinding.For simplicity, we assume that a motor takes either the bound state where both of the ABD are bound to filaments, or the free state where neither of the two ABD are bound to filaments.Unbinding of a motor ABD occurs in three cases.Firstly, a motor ABD that is bound to a filament unbinds stochastically at the rate of ω m .Secondly, when a motor ABD reaches the end of a filament by actively walking along it, it unbinds instantaneously with the probability unity.Thirdly, a motor ABD unbinds with the probability unity when the other ABD of the motor unbinds.The free motors are assumed to diffuse sufficiently fast, so that they distribute uniformly.Then, a free motor binds to a randomly-selected pair of filaments whenever it find a position on each filament that are separated by the length of the motor L m 0 .Thus, this motor binding process alone causes no force on the system.The active force is generated in the cytoskeleton through the motor stretching and bending energies that are stored when motor ABD move along filaments.
Like the motors, we assume that a crosslinker takes either the bound state where both of the ABD are bound to filaments, or the free state where neither of the two ABD are bound to filaments.Unbinding of a crosslinker ABD that is bound to a filament unbinds stochastically at the rate of ω cl , or with the probability unity when the other ABD of the crosslinker unbinds.The free crosslinkers are assumed to diffuse sufficiently fast, so that they distribute uniformly.Then, a free crosslinker binds to a randomly-selected pair of filaments whenever they find a position on each filament that are separated by the length of the crosslinker L cl 0 .Thus, like the motor binding process, this binding process of the crosslinkers alone causes no force on the system.
The third stochastic process is filament turnover that takes place at the rate of ω f .The filament undergoing turnover is placed back into the system immediately at random position with random orientation, and all the motors and crosslinkers previously bound to it become free.
In order to include the effect of tube expansion, we introduce frictional asymmetry that has been proposed in the previous study using active gel model [21].Since our experimental data suggests that the formin actin nucleator protein DAAM plays a key role to align the fused actin nanoclusters and actin stripes in the circumferential direction, we assume that the friction asymmetry affects only actin filaments.Then, Eq. ( S1) is separated to the spatial components as where γ f y /γ f x measures the friction asymmetry.γ f y /γ f x = 1 corresponds to the symmetric case.The set of time-evolution equations is solved numerically in the following manner; First, we calculate the turnover of the filaments, motors, and crosslinkers, and the position of the motor ABD is updated by solving Eq. (S4) with the Euler method.Then, Eqs.(S1)-(S3) are solved by using the fourth-order Runge-Kutta method.In both cases, the time increment is set to dt = 5 × 10 −5 .The simulation results are visualized by using OVITO Pro [22].

II. PARAMETER VALUES
The simulation parameters for the filaments, motors, and crosslinkers are set as summarized in Table S1, unless otherwise stated.Some of the parameters are compared with existing experimental measurements.We consider the dynamics in the membrane vicinity where the diffusion coefficient is about ten times smaller than that in bulk [26], and thus the effective cytosolic viscosity η cyto in the membrane vicinity is assumed about 10 times larger than in the bulk.Therefore, we set η cyto = 10 −1 Pa • sec which is larger than the measured cytosolic viscosity 10 −3 -10 −1 Pa • sec [27].The simulation box is set as 3 µm × 3 µm with periodic boundary conditions in both directions.

III. STEADY-STATE SOLUTIONS
We start the simulation from the initial condition where filaments are randomly placed with random orientation, to which motors and crosslinkers bind randomly following the binding rules.After relaxation that takes about 30-60 sec, we obtained steady-state solutions.By changing parameters including the number of motors and crosslinkers as well as the friction asymmetry, we found that the actin nanoclusters are formed under the existence of sufficient number of crosslinkers without friction asymmetry (Fig. 4b).In contrast, actin stripes, which correspond to the regularly-spaced actin cables found in the experiment, are obtained under the existence of friction asymmetry and sufficient number of motors.See also Supplementary Movie 4. We discuss how these structures are organized in more detail in this section.

III.1. Actin nanoclusters
First, we consider the simplest case with filaments and crosslinkers in the absence of friction asymmetry, in which the actin nanoclusters are formed.One key component for the actin nanocluster formation is the crosslinkers.In fact, if the number of crosslinkers is sufficiently large (N cl = 6750), the filaments organize into clusters as shown in Supplement Figure 7a1.See also the snapshots in Fig. 4b in the main text and Supplementary Movie 4. The size of these clusters is about 200-300 nm, which is comparable to the actin nanoclusters observed in the experiment (Fig. 4c in the main text and Supplement Figure 6b).If the number of crosslinkers is small, however, the filaments do not form nanoclusters but are distributed randomly.
This brings us to a question whether these nanoclusters are also formed under the existence of motors.To check this point, we perform a simulation with a finite number of motors (Supplement Figure 7a2).When the number of motors is small (N m = 100), the nanoclusters are still observed.For a large number of motors (N m = 900), however, the filaments form a labyrinth pattern.From these, we conclude that the motors are not necessary for the nanocluster formation.A small number of motors do not alter the structure, while too many of them change the pattern to a labyrinth.
To further investigate which factor is important for the cluster formation, we focus on the case without motors and vary the length and turnover rate of the crosslinkers as well as those of the filaments.If the turnover rate of the crosslinkers is low (ω cl = 0.001 and 0.01), the filaments organized into nanoclusters (Supplement Figure 7a3).In contrast, for a high turnover rate of the crosslinkers (ω cl = 0.03), no clear clusters are observed and the filaments are distributed randomly.These results indicate that the crosslinkers play an important role in the clustering of filaments.That is, the crosslinkers connect the filaments and tend to keep them close to each other, leading to the effective attraction of filaments.However, this effective attraction of filaments reduces as the crosslinker turnover rate increases, resulting in the random distribution of filaments.
In the case of a rapid filament turnover (ω f = 0.5), the filament distribution becomes rather uniform (Supplement Figure 7a4).As the filament turnover rate decreases, however, nanoclusters appear (ω f = 0.1).Further decreasing ω f makes the nanocluster structure more compact and the number of the clusters decreases, leaving a clear larger space void of filaments between clusters (ω f = 0.01).See also Supplement Figure 6c and Supplementary Table 7.These results suggest that the filament turnover enhances the effective filament diffusion, which acts against the effective attraction to control the distance between the clusters.
We next consider the role of the length of the filaments and crosslinkers.On the one hand, as the crosslinker length increases, the clusters become loose (Supplement Figure 7a5 L cl 0 = 0.04), and eventually no clear clusters are observed (L cl 0 = 0.06).This suggests that the short length of the crosslinkers is essential to keep the filaments close enough to each other.On the other hand, if the filament length is decreased, we still observe clustering of the filaments (Supplement Figure 7a6 L f 0 = 0.075).This result seems counterintuitive as the ratio of the crosslinker length with respect to the filament length becomes large in the case of short filaments.Since the number of filaments is kept constant, however, the total length of the filaments, i.e. the total amount of actin in the system also decreases with the filament length.This is the reason why the clusters are still observed in the case of short filaments (Supplement Figure 7a6 L f 0 = 0.075).This also explains the result of the long filaments, where the clusters start to connect to each other (L f 0 = 0.125).In summary, filaments and a sufficient number of crosslinkers self-organize into nanoclusters in the absence or under the existence of a small number of motors.The results suggest that the clusters are caused by the competition between the effective attraction of the filaments due to the crosslinkers and the effective diffusion of the filaments.The sufficient effective attraction between the filaments connected by crosslinkers enables the filaments being further connected by other crosslinkers if there are more filaments and free crosslinkers in the neighbour.Therefore, a small number of crosslinkers, or high turnover rate of the crosslinkers, both of which decrease the effective attraction, result in a random or uniform filament distribution without any clear pattern.The effective filament diffusion controls the distance between the clusters.A high filament turnover rate enhances the effective diffusion, which disperses the filaments, leaving no cluster pattern.The length of the crosslinkers and the filament density are also important to make the clusters compact and well separated from each other.Finally, a small number of motors does not strongly affect the cluster pattern, although too many of them changes the structure to a labyrinth pattern.

III.2. Actin stripes
Here, we focus on the formation of the stripe pattern.We start from the labyrinth pattern that appears from the filaments mixed with sufficient number of crosslinkers and motors (Supplement Figure 7a2 N m = 900).The labyrinth pattern is a connected structure consisting of actin filaments, crosslinkers and motors without any specific direction, whereas the stripe pattern is extended in the circumferential direction.To provide the external cue, we include friction asymmetry that affects only the filaments following the previous study [21].The results explained in the previous section III.1 correspond to the case where the friction asymmetry γ f y /γ f x is unity.As the friction asymmetry γ f y /γ f x increases from unity, the connected structure of filaments tends to align in the direction of higher friction (Supplement Figure 7b1).For γ f y /γ f x 1.5, we obtained a stripe pattern.
Next, we study the effect of the motors on the stripe pattern formation under the existence of friction asymmetry.Starting from the stripe pattern, we first decrease the number of motors.Then, the connection of the pattern gets worse (Supplement Figure 7b2 N m = 300).By further decreasing the number of motors, the stripe pattern eventually disappears, leaving many discrete clusters that are slightly elongated in the higher friction direction (N m = 100 and 0).See also Figs. 4b (2-M) and 4c and Supplementary Movie 4.
To make more clear the role of the motors, we change the motor walk force.When the motor walk force is absent, the filaments exhibit a connected structure without clear directionality, i.e. a labyrinth pattern (Supplement Figure 7b3 f walk = 0).As the motor activity increases, the structure aligns in the higher friction direction, leading to a stripe pattern (f walk = 0.001 and 0.005).Further increasing motor walk force, the structure become loose and no clear pattern is observed (f walk = 0.05).This is probably because the higher motor walk force adds more energy to the system and enhances the effective diffusion of the filaments, which makes it more difficult to maintain the self-organized structure.
These results suggest that, in the absence of motors or under the existence of small number of motors, the filaments and crosslinkers organize into nanoclusters, which react to the friction asymmetry and tend to elongate in the higher friction direction.Then, as the number of motors increases, these nanoclusters are connected by motors and eventually the pattern becomes stripes aligned in the higher friction direction.This leads to a question whether the stripe pattern appears in the presence of motors and friction asymmetry for the parameter region where the filaments fail to form clusters in the absence of the motors and friction asymmetry.To answer this question, we vary the turnover rate and the length of filaments and crosslinkers, which influence the cluster formation in the absence of motors.
First, we change the turnover rate of crosslinkers and filaments starting from the stripe pattern for ω cl = 0.01 and ω f = 0.1.(Supplement Figure 7b4 and 7b5).On the one hand, for a high turnover rate of crosslinkers or filaments, where no clusters are observed in the absence of friction asymmetry, filaments distribute randomly (Supplement Figure 7b4 ω cl = 0.02 and Supplement Figure 7b5 ω f = 0.2).On the other hand, by decreasing ω cl , the stripe pattern becomes more compact in width (ω cl = 0.005), and by further decreasing ω cl , the connection of the pattern gets worse (ω cl = 0.001).For a lower filament turnover rate, the filaments exhibit a compact stripe pattern (ω f = 0.05).For a very low turnover rate (ω f = 0.01), the filaments form clusters.
Next, we consider the impact of the length of filaments and crosslinkers on the stripe pattern formation.When the length of crosslinkers or filaments are increased, the stripe structure becomes loose (Supplement Figure 7b6  and 7b7).For a shorter filament length, however, the stripe structure gets more compact (Supplement Figure 7b6 L f 0 = 0.075).For very short filaments, many small clusters are formed that are connected by motors (Supplement Figure 7b7 L f 0 = 0.05).This is probably due to the decrease in filament density, i.e. the decrease in the total amount of actin molecules.In contrast, for a shorter crosslinker length, the stripe structure again becomes loose (Supplement Figure 7b6 L cl 0 = 0.03).This may be because the crosslinkers are now too short to connect filaments.To summarize, the actin nanoclusters, that appear as a result of the competition between the effective diffusion and the effective attraction acting between the filaments due to crosslinkers, react to the friction asymmetry and tend to elongate in the higher friction direction.Then, the motors connect these elongated actin nanoclusters and form stripes that align in the direction of higher friction.Therefore, the motors and the friction asymmetry are essential for the transition from the clusters to the stripe pattern.In fact, in the absence of the friction asymmetry but in the presence of motors, the structure loses preferred direction and the labyrinth pattern appears.In the absence or under the existence of small number of motors, the filaments form discrete clusters.In addition, the balance between the effective attraction of filaments due to crosslinkers and their effective diffusion is important for the stripe formation.On the one hand, on the conditions where the filaments fail to form clusters in the absence of motors and friction asymmetry, the stripe pattern is not obtained even in the presence of motors and friction asymmetry.On the other hand, for the parameters where the filaments form very compact clusters in the absence of motors and friction asymmetry, the clusters still remain in the presence of motors and friction asymmetry, and the transition to the stripe pattern does not occur.

Supplementary Figure 1 :Supplementary Figure 2 :
Cluster-to-cable transition of cortical actin pattern in the expanding tracheal tube.(a) The actin pattern visualized by lifeact::GFP in the fixed embryos at stages 14 and 17, respectively.The lower square images indicate a magnified view of the yellow square and its binarized image.Three experiments were repeated independently.Scale bars: 2 µm (upper) and 0.5 µm (lower).(b, c) Comparison of the cluster-fitted ellipses between stage 15 (n=356, N=22) and stage 16 (n=271, N=21) with length of the major axis (b) and length of the minor axis (c).For statistical analyses, twosided two-sample Kolmogorov-Smirnov test was performed.p=0.00981(b) and p=0.329 (c).**p < 0.01.See Methods' Cluster analysis for detail.(d) Average interval of the periodic circumferential cables at early stage 16.Boxplots represent median plus minima and maxima with lower and upper quantiles.The number of cables (n) and embryos (N) counted for the measurement are indicated in the plot.(e) The actin pattern visualized by Phalloidin555, Utrophin::GFP, and lifeact::mScarletx2 in the fixed embryos at stages 15 and 16, respectively.The lower square images indicate magnified views.Scale bars: 2 µm (upper) and 0.5 µm (lower).(f) The mean intensities of RFP::RhoGEF2 at vicinity of longitudinal junctions or circumferential junctions are measured and those ratio are plotted.The accumulation of RhoGEF2 to the longitudinal junction that is seen at larval stage was not observed at embryonic stages 15 nor 16.Boxplots represent median plus minima and maxima with lower and upper quantiles.(g) The transition of average skeleton length (green line) and S.D. (light grey lines) are indicated.n=6 (-80, -70, -60, -50, -40 min), 7 (-30 min), 9 (-20, -10, 0 min), 8 (10 min), or 5 (20 min) biologically independent samples.(h) The transverse (yz) images of the tracheal tube during expansion period, stage 15.The time point of circumferential cables formed was defined as 0 min, and the relative time points are written above.Scale bar, 1 µm.Detection of the nanocluster fusion sites and measurement of the duration.(a) The potential fusion sites of the actin nanoclusters at time point tn are selected by

TABLE S1 .
The summary of the simulation parameters.The ones indicated in bold font are the input parameters of the simulation.